Apparently this is a multiple choice question, judging from your other posts (which make it clearer).
But still, the wording is a bit puzzling.
By domain of Q, do mean the extension of Q?
If so, then what is the domain under discussion? The set of the first three natural numbers, {0,1,2}?
If so, and we write out the expansion of that existentially quantified formula, it should look like:
~Q(0,0,0) V ~Q(0,0,1) V ~Q(0,0,2)
Then clearly, none of the four will fill the bill.
But of course, I may be completely misreading the question; it won't be the first time.
Sorry, I have to go. I can look back later, but maybe before that someone else will help.
The answer to #4 is:
TFT
Ax x+1 > x
is obviously true, where 'x' ranges over real numbers.
Ax 2x = 3x, is false where 'x' ranges over real numbers, since, e.g. 2*1 does not equal 3*1.
Ex x = -x, is true where 'x' ranges over real numbers, since 0 = -0, so there is at least one real number x for which x = -x is true.
/
The answer to #3 is:
a.
You're being asked to translate Ay Q(0 y 0) into a conjunction or disjunction. But the only values allowed for 'y' are 0 or 1.
So for Q(0 y 0) to hold for all allowed values of 'y' is to say that it holds for both 0 and for 1, when 0 or 1, respectively replace 'y'.
So, in these circumstances,
Ay Q(0 y 0)
becomes
Q(0 0 0) /\ Q(0 1 0)
That's all there is to it, really.
Hello Rozana
is an example of a propositional function; that is, a function that returns a truth value (i.e. the value True or False) that will depend upon the value(s) of any parameter(s) that are supplied to it.
Here, the value of will be determined by the values of three parameters, and . The values that can take are given in the question - defined as the domain of .
Since can take different values ; and can take different values ; and can also take values , there are possible propositions that can be represented by , each one returning the value True or False. They are:Without knowing the actual details of what is, we can't say what value each of these propositions has. However, we can say, that if (as in your original question) is a true statement, then for one or both of the possible values of ( and ), is true. In other words, is true or is true (or both). So:
can be written as****
Question 3 can be answered in the same way:meansFor all (that is, and ), is true.So
is true and is true.Therefore:
can be written as****
Question 4
Examine the truth of each of the propositions in turn:is True. It simply says that if we add to any real number, the answer is greater than the number we started with.Clearly is False. To show this, we only need find one value of for which . (That's not very hard, is it?)
is True. There is a value of for which ; that is .
So the ordered triple has the truth value .
Does that clear up the problems?
Grandad